Average Error: 29.5 → 1.0
Time: 3.2m
Precision: 64
Internal Precision: 1344
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le 26.35935858304994:\\ \;\;\;\;\frac{(\left(\left(x \cdot x\right) \cdot \frac{2}{3}\right) \cdot x + \left(2 - x \cdot x\right))_*}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{1}{\varepsilon} + 1\right) \cdot e^{\left(-x\right) \cdot \left(1 - \varepsilon\right)} - \left(\frac{1}{\varepsilon} - 1\right) \cdot {e}^{\left(\left(1 + \varepsilon\right) \cdot \left(-x\right)\right)}}{2}\\ \end{array}\]

Error

Bits error versus x

Bits error versus eps

Derivation

  1. Split input into 2 regimes

  2. if x < 26.35935858304994

    1. Initial program 39.0

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]

    2. Taylor expanded around 0 1.2

      \[\leadsto \frac{\color{blue}{\left(\frac{2}{3} \cdot {x}^{3} + 2\right) - {x}^{2}}}{2}\]
    3. Using strategy rm

    4. Applied add-cube-cbrt1.2

      \[\leadsto \frac{\left(\frac{2}{3} \cdot {x}^{3} + 2\right) - \color{blue}{\left(\sqrt[3]{{x}^{2}} \cdot \sqrt[3]{{x}^{2}}\right) \cdot \sqrt[3]{{x}^{2}}}}{2}\]

    5. Applied add-cube-cbrt2.7

      \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{\frac{2}{3} \cdot {x}^{3} + 2} \cdot \sqrt[3]{\frac{2}{3} \cdot {x}^{3} + 2}\right) \cdot \sqrt[3]{\frac{2}{3} \cdot {x}^{3} + 2}} - \left(\sqrt[3]{{x}^{2}} \cdot \sqrt[3]{{x}^{2}}\right) \cdot \sqrt[3]{{x}^{2}}}{2}\]

    6. Applied prod-diff2.7

      \[\leadsto \frac{\color{blue}{(\left(\sqrt[3]{\frac{2}{3} \cdot {x}^{3} + 2} \cdot \sqrt[3]{\frac{2}{3} \cdot {x}^{3} + 2}\right) \cdot \left(\sqrt[3]{\frac{2}{3} \cdot {x}^{3} + 2}\right) + \left(-\sqrt[3]{{x}^{2}} \cdot \left(\sqrt[3]{{x}^{2}} \cdot \sqrt[3]{{x}^{2}}\right)\right))_* + (\left(-\sqrt[3]{{x}^{2}}\right) \cdot \left(\sqrt[3]{{x}^{2}} \cdot \sqrt[3]{{x}^{2}}\right) + \left(\sqrt[3]{{x}^{2}} \cdot \left(\sqrt[3]{{x}^{2}} \cdot \sqrt[3]{{x}^{2}}\right)\right))_*}}{2}\]

    7. Applied simplify1.2

      \[\leadsto \frac{\color{blue}{(\left(\left(x \cdot x\right) \cdot \frac{2}{3}\right) \cdot x + \left(2 - x \cdot x\right))_*} + (\left(-\sqrt[3]{{x}^{2}}\right) \cdot \left(\sqrt[3]{{x}^{2}} \cdot \sqrt[3]{{x}^{2}}\right) + \left(\sqrt[3]{{x}^{2}} \cdot \left(\sqrt[3]{{x}^{2}} \cdot \sqrt[3]{{x}^{2}}\right)\right))_*}{2}\]

    8. Applied simplify1.2

      \[\leadsto \frac{(\left(\left(x \cdot x\right) \cdot \frac{2}{3}\right) \cdot x + \left(2 - x \cdot x\right))_* + \color{blue}{0}}{2}\]

    if 26.35935858304994 < x

    1. Initial program 0.4

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
    2. Using strategy rm

    3. Applied *-un-lft-identity0.4

      \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{\color{blue}{1 \cdot \left(-\left(1 + \varepsilon\right) \cdot x\right)}}}{2}\]

    4. Applied exp-prod0.4

      \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot \color{blue}{{\left(e^{1}\right)}^{\left(-\left(1 + \varepsilon\right) \cdot x\right)}}}{2}\]

    5. Applied simplify0.4

      \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot {\color{blue}{e}}^{\left(-\left(1 + \varepsilon\right) \cdot x\right)}}{2}\]
  3. Recombined 2 regimes into one program.

  4. Applied simplify1.0

    \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;x \le 26.35935858304994:\\ \;\;\;\;\frac{(\left(\left(x \cdot x\right) \cdot \frac{2}{3}\right) \cdot x + \left(2 - x \cdot x\right))_*}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{1}{\varepsilon} + 1\right) \cdot e^{\left(-x\right) \cdot \left(1 - \varepsilon\right)} - \left(\frac{1}{\varepsilon} - 1\right) \cdot {e}^{\left(\left(1 + \varepsilon\right) \cdot \left(-x\right)\right)}}{2}\\ \end{array}}\]

Runtime

Time bar (total: 3.2m)Debug logProfile

herbie shell --seed 2019053 +o rules:numerics
(FPCore (x eps)
  :name "NMSE Section 6.1 mentioned, A"
  (/ (- (* (+ 1 (/ 1 eps)) (exp (- (* (- 1 eps) x)))) (* (- (/ 1 eps) 1) (exp (- (* (+ 1 eps) x))))) 2))